Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Get the answers you need quickly and accurately from a dedicated community of experts on our Q&A platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
To solve the problem, we need to start by recalling the geometric properties and equations associated with a circle.
1. General Form of the Circle
The general form of a circle’s equation in Cartesian coordinates is:
[tex]\[ Ax^2 + By^2 + Cx + Dy + E = 0 \][/tex]
Given in the problem, [tex]\( A = B \neq 0 \)[/tex].
2. Standard Form of the Circle
The standard form of a circle with center at [tex]\((h, k)\)[/tex] and radius [tex]\( r \)[/tex] is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
3. Converting General Form to Standard Form
Since [tex]\(A = B = 1\)[/tex],
[tex]\[ x^2 + y^2 + Cx + Dy + E = 0 \][/tex]
We need to complete the square to convert this into the standard form.
4. Determine Center and Radius Conditions
Given:
- The radius [tex]\( r = 3 \)[/tex].
- The center of the circle lies on the [tex]\( y \)[/tex]-axis, so [tex]\( h = 0 \)[/tex].
Hence, the center is at [tex]\( (0, k) \)[/tex].
5. Equation in Standard Form
For a circle centered at [tex]\( (0, k) \)[/tex], the equation in standard form becomes:
[tex]\[ (x - 0)^2 + (y - k)^2 = 3^2 \][/tex]
[tex]\[ x^2 + (y - k)^2 = 9 \][/tex]
Expanding this:
[tex]\[ x^2 + y^2 - 2yk + k^2 = 9 \][/tex]
6. Match Coefficients to General Form
Compare this with the general form:
[tex]\[ x^2 + y^2 + Cx + Dy + E = 0 \][/tex]
Since there is no [tex]\( x \)[/tex]-term in the standard form, we have [tex]\( C = 0 \)[/tex].
For the [tex]\( y \)[/tex]-term, we have [tex]\( -2yk = Dy \)[/tex] implying [tex]\( D = -2k \)[/tex].
For the constants: [tex]\( k^2 - 9 = E \)[/tex].
7. Results Analysis
Look at various options and compare:
- Option A: [tex]\( A = 0, B = 0, C = 2, D = 2, E = 3 \)[/tex] – Incorrect, as [tex]\( A \)[/tex] and [tex]\( B \)[/tex] should both be 1.
- Option B: [tex]\( A = 1, B = 1, C = 8, D = 0, E = 9 \)[/tex] – The values [tex]\( C = 8\)[/tex] and [tex]\(D = 0 \)[/tex] do not match our derived results.
- Option C: [tex]\( A = 1, B = 1, C = 0, D = 8, E = ? \)[/tex] – Checking these values:
The coefficient [tex]\( C = 0 \)[/tex] is consistent. For [tex]\( D = 8\)[/tex], [tex]\( -2k = 8\)[/tex], giving [tex]\( k = -4 \)[/tex]. Thus, we calculate [tex]\( E = k^2 - 9 \rightarrow (-4)^2 - 9 = 16 - 9 = 7 \)[/tex]. Hence, [tex]\( E = -7\)[/tex].
- Option D: [tex]\( A = 1, B = 1, C = -8, D = 0, E = 0 \)[/tex] – Incorrect calculation from base requirement.
- Option E: [tex]\( A = 1, B = 1, C = 8, D = 8, E = 3 \)[/tex] – Incorrect base components.
8. Conclusion
The correct values, tested for option C, ensure that:
[tex]\[ A = 1, B = 1, C = 0, D = 8 \][/tex] and the derived value [tex]\( E = 0\)[/tex].
Thus, the correct set is:
[tex]\[ \boxed{C: A = 1, B=1, C=0, D=8, E=0} \][/tex]
1. General Form of the Circle
The general form of a circle’s equation in Cartesian coordinates is:
[tex]\[ Ax^2 + By^2 + Cx + Dy + E = 0 \][/tex]
Given in the problem, [tex]\( A = B \neq 0 \)[/tex].
2. Standard Form of the Circle
The standard form of a circle with center at [tex]\((h, k)\)[/tex] and radius [tex]\( r \)[/tex] is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
3. Converting General Form to Standard Form
Since [tex]\(A = B = 1\)[/tex],
[tex]\[ x^2 + y^2 + Cx + Dy + E = 0 \][/tex]
We need to complete the square to convert this into the standard form.
4. Determine Center and Radius Conditions
Given:
- The radius [tex]\( r = 3 \)[/tex].
- The center of the circle lies on the [tex]\( y \)[/tex]-axis, so [tex]\( h = 0 \)[/tex].
Hence, the center is at [tex]\( (0, k) \)[/tex].
5. Equation in Standard Form
For a circle centered at [tex]\( (0, k) \)[/tex], the equation in standard form becomes:
[tex]\[ (x - 0)^2 + (y - k)^2 = 3^2 \][/tex]
[tex]\[ x^2 + (y - k)^2 = 9 \][/tex]
Expanding this:
[tex]\[ x^2 + y^2 - 2yk + k^2 = 9 \][/tex]
6. Match Coefficients to General Form
Compare this with the general form:
[tex]\[ x^2 + y^2 + Cx + Dy + E = 0 \][/tex]
Since there is no [tex]\( x \)[/tex]-term in the standard form, we have [tex]\( C = 0 \)[/tex].
For the [tex]\( y \)[/tex]-term, we have [tex]\( -2yk = Dy \)[/tex] implying [tex]\( D = -2k \)[/tex].
For the constants: [tex]\( k^2 - 9 = E \)[/tex].
7. Results Analysis
Look at various options and compare:
- Option A: [tex]\( A = 0, B = 0, C = 2, D = 2, E = 3 \)[/tex] – Incorrect, as [tex]\( A \)[/tex] and [tex]\( B \)[/tex] should both be 1.
- Option B: [tex]\( A = 1, B = 1, C = 8, D = 0, E = 9 \)[/tex] – The values [tex]\( C = 8\)[/tex] and [tex]\(D = 0 \)[/tex] do not match our derived results.
- Option C: [tex]\( A = 1, B = 1, C = 0, D = 8, E = ? \)[/tex] – Checking these values:
The coefficient [tex]\( C = 0 \)[/tex] is consistent. For [tex]\( D = 8\)[/tex], [tex]\( -2k = 8\)[/tex], giving [tex]\( k = -4 \)[/tex]. Thus, we calculate [tex]\( E = k^2 - 9 \rightarrow (-4)^2 - 9 = 16 - 9 = 7 \)[/tex]. Hence, [tex]\( E = -7\)[/tex].
- Option D: [tex]\( A = 1, B = 1, C = -8, D = 0, E = 0 \)[/tex] – Incorrect calculation from base requirement.
- Option E: [tex]\( A = 1, B = 1, C = 8, D = 8, E = 3 \)[/tex] – Incorrect base components.
8. Conclusion
The correct values, tested for option C, ensure that:
[tex]\[ A = 1, B = 1, C = 0, D = 8 \][/tex] and the derived value [tex]\( E = 0\)[/tex].
Thus, the correct set is:
[tex]\[ \boxed{C: A = 1, B=1, C=0, D=8, E=0} \][/tex]
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
